Problem: Find the gradient of $f(x, y) = xe^y$ at $(4, 0)$. $\nabla f(4, 0) = ($
The gradient of a scalar field is all its partial derivatives put together into a vector. For a 2D scalar field, this looks like $\nabla f = (f_x, f_y)$. Let's find $f_x$ and $f_y$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ xe^y \right] \\ \\ &= e^y \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ xe^y \right] \\ \\ &= xe^y \end{aligned}$ Now we can evaluate the partial derivatives we found at the point $(4, 0)$. $\begin{aligned} f_x(4, 0) &= e^y = e^0 = 1 \\ \\ f_y(4, 0) &= xe^y = 4e^0 = 4 \end{aligned}$ The gradient of $f$ at $(4, 0)$ is $\nabla f = (1, 4)$.